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# Notes03 - MA3245 Financial Mathematics I Notes 3 Basic...

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MA3245: Financial Mathematics I Notes 3: Basic option theory (b) Dr. Oliver Chen [email protected] Department of Mathematics, National University of Singapore

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1 Introduction 2 American options 3 Interest rates 4 Summary MA3245 Notes 3: Basics (b) 2 / 52
Introduction Arbitrage argument for European options Arbitrage argument for European options For European options, exercise can only occur at maturity so that arbitrage arguments are relatively simple. As an exercise, prove the following: Proposition: Suppose European derivatives A and B have a common maturity T and common underlying S t . Suppose further that their prices are V A , t and V B , t , and their payoffs are f A ( S T ) and f B ( S T ). If f A ( S T ) f B ( S T ) S T then V A , t V B , t MA3245 Notes 3: Basics (b) 4 / 52

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Introduction Arbitrage argument for European options Arbitrage for Europeans (cont.) An easy corollary of the last proposition: Corollary: With the same notation, if f A ( S T ) = f B ( S T ) S T , then V A , t = V B , t Arbitrage arguments are not as simple when exercise is allowed before maturity. In that case, we will have to take into consideration the option holder’s rights before maturity as well. MA3245 Notes 3: Basics (b) 5 / 52
Introduction Exotic options Exotic options European calls and puts are sometimes referred to as plain vanilla options. All others are referred to as exotic options. European options where only the price at maturity matters are path-independent options. MA3245 Notes 3: Basics (b) 6 / 52

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Introduction Exotic options Path dependent options Asian options depend on some sort of average. Suppose Δ t is a time interval. An Asian rate call option with maturity n Δ t can be defined as a European option, except that some average is used for the asset value. The average can be arithmetic, so that the payoff at maturity is: 1 n n i =1 S i Δ t - K + MA3245 Notes 3: Basics (b) 7 / 52
Introduction Exotic options Path dependent options (cont.) Or the average can be geometric, so that the payoff at maturity is: [ S Δ t S t · · · S n Δ t ] 1 / n - K + A lookback option depends on the ‘optimal’ value over the lifetime of the option. For example, a floating strike lookback put has a payoff: max 0 t T S t - S T MA3245 Notes 3: Basics (b) 8 / 52

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Introduction Exotic options Path dependent options (cont.) Barrier options are another type of path dependent options. Single barrier options come in two flavours: knock-in and knock-out. Knock-in options are worthless unless the asset value goes through a preset barrier, in which case they become a European option. Knock-out options are worthless if the asset value goes through a preset barrier. If they do not go through a barrier, then they just remain a European option. MA3245 Notes 3: Basics (b) 9 / 52
Introduction Exotic options Barrier option example Suppose S 0 = \$100.

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